When measuring the compression efficiency of encodings, it is natural to assume that
\(\sigma \)  is a non-decreasing function of \(n\) .
In such premises, if each \(d_j\)  component occupies \(\lceil \log n\rceil \)  bits and
each \(\ell _j\)  component takes \(O(1 + \log \ell _j)\)  bits, then it is known
that the size of the LZ encoding is upperbounded by \(nH_k + o(n\log \sigma )\)  bits,
provided \(k\)  is a function of \(n\)  such that
\(k = o(\log _\sigma n)\) ; see [1]}, [2]}, [3]}.
In the sequel we also utilize a slightly different encoding that, for each \(d_j\) , uses
a universal code [4]}, [5]} taking \(\log d_j + O(1 + \log \log d_j)\)  bits.
